I'll likely be without internet a little over a month from now, due to moving. I don't want to break my (currently) 636 consecutive days visiting SciFi.SE, so I was checking my phone to make sure I could access the site just fine.
I logged in and even ended up VTCing a question. But the "last seen" field on my profile did not update until I loaded the full site on my laptop.
Aside from that, this would seem to indicate that the "Enthusiast" and "Fanatic" badges are unavailable to mobile-only users.